Study Notes on Properties of Gases

Unit 1: Properties of Gases Part B

Faculty of Science Department of Chemistry Physical Chemistry IIIA PCA216X Dr Maphoru MV

Learning Outcomes

• Defining Effusion and Diffusion:

  • Effusion is the escape of gas molecules through a tiny hole into an evacuated space (e.g., a punctured tyre).

  • Diffusion is the spread of one substance throughout another substance or space (e.g., perfume diffusing throughout a room).

• Deriving Graham’s Law of Effusion:

  • Use Graham’s Law in calculations regarding effusion rates.

• Kinetic Molecular Theory (KMT):

  • Discuss and explain the behavior of an ideal gas using KMT principles.

• Effect of Temperature on Molecular Speeds:

  • Discuss how temperature affects the distribution of molecular speeds of gases.

• Defining Root-Mean-Square Speed:

  • Define and calculate the root-mean-square (rms) speed of a gas.

Graham's Law of Effusion

Definitions

• Effusion:

  • The escape of gas molecules through a tiny hole into an evacuated space (Example: a punctured tyre).

• Diffusion:

  • The process where one substance spreads throughout a second substance or space (Example: the scent of perfume spreading throughout a room).

Graham's Law

• Statement of the Law:

  • The rate of effusion of a gas is inversely proportional to the square roots of their densities when compared at identical pressures and temperatures:
    $r ext{ (effusion rate)} ext{ is inversely proportional to } rac{1}{ ext{molar mass}}$

  • If the rates of effusion for two gases are denoted as $r<em>1$ and $r</em>2$ and their respective molar masses as $M<em>1$ and $M</em>2$, then:
    $rac{r<em>1}{r</em>2} = rac{ ext{sqrt}(M<em>2)}{ ext{sqrt}(M</em>1)}$

Relationships and Rearranging

• In constant temperature and pressure conditions,
  $r<em>1 imes d</em>1 = r<em>2 imes d</em>2$
  where $d$ represents the density.

• Density of a gas is directly proportional to its molecular mass, therefore:
  $rac{r<em>1}{r</em>2} = rac{M<em>2}{M</em>1}$

Tutorials

Tutorial 1

• Objective: Determine which gas among several will effuse at the highest rate when kept at the same temperature and pressure. Justify the answer.

Problem Set for Tutorial 1

1. Given that ammonia effuses at a rate 2.93 times that of an unknown gas, calculate the molecular mass of the unknown gas.

2. Tutorial 2 & 3: Determine the gas that will effuse rapidly and the ratio of effusion rates based on the density:

   • The density of $CO<em>2$ is 1.96 g/L and that of $N</em>2$ is 1.25 g/L.

   • Determine the gas that will effuse rapidly, using the density to make calculations.

Kinetic Molecular Theory (KMT)

Overview of KMT

• The Ideal Gas Laws describe gas behavior but do not explain the reasoning behind their behavior.

• KMT provides a model to understand the physical properties of gases based on several hypotheses.

Hypotheses of KMT

1. Gases consist of individual particles (atoms or molecules) that are significantly smaller than the distances between them.

2. Particles are in constant, random motion, possessing kinetic energy.

3. No attractive or repulsive forces exist between the particles.

4. The volume of gas molecules is negligible compared to the total volume of the gas.

5. Energy can be transferred during collisions between molecules, but their average kinetic energy remains constant at a constant temperature (Elastic collisions).

6. The average kinetic energy of gas molecules is directly proportional to the absolute temperature:
   $KE ext{ (kinetic energy)} ext{ is proportional to T}$

Average Kinetic Energy Calculations

• Describes how pressure develops in a container based on gas molecular collisions.

Pressure Relations and Kinetic Theory

Average Kinetic Energy Derivation

• Momentum change from gas molecules colliding elastically with a container:

  • Before collision: $p<em>{1x} = mu</em>{1x}$

  • After collision: $p<em>{1x} = -mu</em>{1x}$

  • Change in momentum: $riangle P = p<em>{f} - p</em>{i} = mu<em>{1x} - (-mu</em>{1x}) = 2mu_{1x}$

• Pressure calculation for one gas molecule colliding with a container:
  $P<em>1 = rac{F</em>1}{A} = rac{mu_{1x}^2}{OB(BC)(AB)}$
  (Volume of container, V = OB x BC x AB)

Pressure of Gas in Directions

• Overall gas pressure is calculated as: $P = rac{1}{3} rac{Nm u^2}{V}$

  • Relating average kinetic energy to the pressure and the number of molecules.

Root Mean Square Velocity (urms)

Derivation and Equation

• Set relationship of average kinetic energy and molar mass to: $KE = rac{1}{2} mu^2$ $or rac{3}{2} k_{B} T$

  • Conservative forms yield:
    $u^2_{rms} = rac{3RT}{M}$

• Interpretation: The rms speed varies with changes in molar mass and temperature:
  $urms = rac{3RT}{M}$

Physical Implications

• The rms speed is inversely proportional to molar mass and directly proportional to the square root of absolute temperature. This results in heavier gases having lower speeds at the same temperature compared to lighter gases.

Kinetic Theory Laws Applications

Boyle's Law

• Describes the relationship of volume and pressure of a gas at constant temperature:
  $V ext{ is inversely proportional to } P$

• Confirmed through molecular collisions, where reducing volume increases collision frequency, thus raising pressure.

Gay-Lussac's Law

• Dictates that pressure is directly proportional to absolute temperature at constant volume:
  $P ext{ is proportional to } T$

• Results from increased average molecule velocity leading to increased collision intensity.

Charles’ Law

• Explains that volume increases with temperature at constant pressure:
  $V ext{ is proportional to } T$

• Allows gas expansion; higher temperature leads to an increase in average kinetic energy and subsequently volume at constant pressure.

Dalton's Law

• Total pressure as the sum of individual partial pressures of gases in a mixture:
  $P<em>{total} = P</em>1 + P<em>2 + P</em>3 + …$

• Each gas particle acts independently in accordance to KMT hypotheses.

Graham's Law Reiteration

• Rate of effusion linked to molar masses as previously established, demonstrating the relationship in practical applications.

Ethical, Philosophical, and Practical Implications of Gas Theory

Absolute Zero

• Defined by the kinetic theory as the temperature at which average kinetic energy approaches zero:
  $KE ext{ correlates to T through average energy measures}$

• Philosophically, this establishes absolute zero as a limit point in thermodynamics.

Practical Applications and Numerical Problem Sets

Tutorial Problem Set Examples

1. Calculate the average kinetic energy at a specific temperature for nitrogen gas in a given scenario.

2. Convert root mean square velocities into temperature measurements for different gases.

3. Rank various samples of gas with varying amounts and temperatures based on speed of atoms or molecules.

4. Further differentiate based on rms values among gases such as helium and xenon, analyzing velocities and implications of ideal gas behavior.